Show that if $(Y_{n}-X_{n})\stackrel{p}{\to}0$ and $F_{X_{n}}\to F_{x}$ then $F_{Y_{n}}\to F_{x}$

Question

I have to prove the following statement:

Let $X_{n}$ and $Y_{n}$ be sequences of random variables with cdf $F_{X_{n}}$ and $F_{Y_{n}}$ respectively, show that if $X_{n}-Y_{n}\stackrel{p}{\to}0$ and $F_{X_{n}}\to F_{X}$ then $F_{Y_{n}}\to F_{X}$

From convergence in probability I know that for every $\epsilon$ there exist $N$ such that

$P\left(\left|X_{n}-Y_{n}\right|>\epsilon\right)=0 $

for $n>N$. I think I should combine this statement with the fact that $X_{n}$ converges in distribution in order to get that $\left|F_{Y_{n}}(y)-F_{X}(y)\right|<\epsilon$ for $n>N$ but cannot make it work. I would appreciate any help

Answer 1

$Y_n \leq x$ implies that either $|Y_n-X_n| >\epsilon$ or $X_n \leq x+\epsilon$. Hence $F_{Y_n}(x) \leq F_{X_n}(x+\epsilon)+P(|X_n-Y_n| >\epsilon)$. Can you conclude from this that $\lim \sup F_{Y_n}(x) \leq F(x)$? For the other direction use the fact that $X_n \leq x-\epsilon$ implies that either $|Y_n-X_n| >\epsilon$ or $Y_n \leq x$ which gives $\lim \inf F_{Y_n}(x) \geq F(x)$ provided $x$ is a continuity point of $F_X$ .